Weak derivative of a locally integrable function

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, $n\ge 1$, let $u,v\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$, and let $\alpha \in {\mathbb{N}}_0^n$ be the multi-index of $C^k$ maps and multi-index derivative notation in Euclidean space. We say that $v$ is the weak derivative $D^{\alpha }u$ if ${\int }_{\mathrm{\Omega }}u{D}^{\alpha }\phi dx=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{\mathrm{\Omega }}v\phi dx\text{for every }\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })\mathrm{.}$ Both integrals are finite because the test and all its derivatives are bounded and compactly supported. The test identity is choice-free. By Locally integrable functions as regular distributions and the signed transpose convention in Distributional derivative, it is equivalent to ${\mathrm{\partial }}^{\alpha }{T}_u=T_v$ as distributions. For $\alpha =0$, this identity reduces to $T_u=T_v$, hence to $u=v$ almost everywhere by the published embedding theorem under The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

Well-definedness. Changing either representative on a null set leaves each test integral unchanged. Weak derivatives are therefore statements about almost-everywhere classes; existence and uniqueness of their locally integrable value classes are audited separately.